Structural Complexity I

Balcázar, José Luís; Dı́az, Josep; Gabarró, Joaquim

doi:10.1007/978-3-642-79235-9

Cited by 171 publications

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“…We use standard complexity theoretic notation as in Balcázar et al (1995) and Papadimitriou (1994). For background and notation in Kolmogorov complexity cc 14 (2005) Language compression and pseudorandom generators 235 we refer the reader to Li & Vitányi (1997).…”

Section: Preliminariesmentioning

confidence: 99%

Language compression and pseudorandom generators

2005

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Abstract. The language compression problem asks for succinct descriptions of the strings in a language A such that the strings can be efficiently recovered from their description when given a membership oracle for A. We study randomized and nondeterministic decompression schemes and investigate how close we can get to the information theoretic lower bound of log A =n for the description length of strings of length n. Using nondeterminism alone, we can achieve the information theoretic lower bound up to an additive term of O(( log A =n + log n) log n); using both nondeterminism and randomness, we can make do with an excess term of O(log 3 n). With randomness alone, we show a lower bound of n − log A =n − O(log n) on the description length of strings in A of length n, and a lower bound of 2 · log A =n − O(1) on the length of any program that distinguishes a given string of length n in A from any other string. The latter lower bound is tight up to an additive term of O(log n). The key ingredient for our upper bounds is the relativizable hardness versus randomness tradeoffs based on the Nisan-Wigderson pseudorandom generator construction.

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Section: Preliminariesmentioning

confidence: 99%

Language compression and pseudorandom generators

2005

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“…Let opt(I 1 ) and opt(I 2 ) be the optimal values of the success ratio function of the instances I 1 and I 2 . Remember that it is trivially true that 4 1 opt(I 1 ) 1 − (1/k). Hence, because of (i),…”

Section: Theorem 3 (Kann Et Al [21]) It Is Np-hard To Approximate Tmentioning

confidence: 99%

On Approximate Learning by Multi-layered Feedforward Circuits

DasGuptaBhaskar

HammerBarbara

2000

Lecture Notes in Computer Science

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We deal with the problem of efficient learning of feedforward neural networks. First, we consider the objective to maximize the ratio of correctly classified points compared to the size of the training set. We show that it is NP-hard to approximate the ratio within some constant relative error if architectures with varying input dimension, one hidden layer, and two hidden neurons are considered where the activation function in the hidden layer is the sigmoid function, and the situation of epsilon-separation is assumed, or the activation function is the semilinear function. For single hidden layer threshold networks with varying input dimension and n hidden neurons, approximation within a relative error depending on n is NP-hard even if restricted to situations where the number of examples is limited with respect to n.Afterwards, we consider the objective to minimize the failure ratio in the presence of misclassification errors. We show that it is NP-hard to approximate the failure ratio within any positive constant for a multilayered threshold network with varying input dimension and a fixed number of neurons in the hidden layer if the thresholds of the neurons in the first hidden layer are zero. Furthermore, even obtaining weak approximations is almost NP-hard in the same situation.

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“…The analogous proposition relating G ω b (M, M , s, s ) and the full modal language does not hold, as is witnessed by the models in Figure 7. 3.…”

Section: Gamesmentioning

confidence: 99%

“…Completeness and hardness are understood in this chapter by means of polynomial-time many-one reductions. (The reader is referred to [3] for basic definitions.) In this section, we concentrate on the satisfiability problem -recall that C-completeness of the satisfiability problem for a modal system implies co-C-completeness of the validity problem.…”

Section: Modal Complexitymentioning

confidence: 99%

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Local Variations on a Loose Theme: Modal Logic and Decidability

Marx

Venema

Finite Model Theory and Its Applications

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This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of models, to determine whether a given formula ϕ is satisfiable in some model of that class (a more precise definition will follow). The general picture is that modal logic behaves quite well in this respect. In fact, many authors follow Vardi [58] in calling modal logic robustly decidable on the ground that most of the nice computational properties of modal logic are preserved if one considers extensions or variants of the basic system. The main aim of this chapter is to refine and analyze this picture.To start with, we should clarify what we are talking about when using the term "modal logic". Traditionally, propositional modal logic would be described as an extension of propositional logic with operators 2 and 3 for talking about the necessity and possibility of a formula being true. However, nowadays the term "modal logic" is used for a plethora of formalisms, with applications in various disciplines ranging from linguistics to economics, see [11,17,39,56] for a sample of applications in computer science.And while (propositional) modal logics will usually still be an extension of classical propositional logic with a number of modal operators, the intended meanings of these operators differ enormously. For instance, the formula 2 a ϕ could mean "player a knows that ϕ is the case" in a formalization of game theory, or "after the execution of program a, ϕ will be the case" in a formal language for program verification. Fortunately, on a technical level, all these formalisms still have a lot in common. That is why this chapter first introduces the notion of a modal system as a triple consisting of a (propositional) modal language, a class of models and a truth function. This definition covers most of the systems that appear in the literature under the name "modal logic"; in particular, the familiar system of basic modal logic, to be discussed in section 7.3.

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Structural Complexity I

Cited by 171 publications

References 0 publications

Language compression and pseudorandom generators

Language compression and pseudorandom generators

On Approximate Learning by Multi-layered Feedforward Circuits

Local Variations on a Loose Theme: Modal Logic and Decidability

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